On the number of (n)-dimensional invariant spheres in polynomial vector fields of (mathbb{C}^{n+1})

We study the polynomial vector fields (mathcal{X}= displaystyle sum_{i=1}^{n+1} P_i(x_1,ldots,x_{n+1}) frac{partial}{partial x_i}) in (mathbb{C}^{n+1}) with (ngeq 1) . Let (m_i) be the degree of the polynomial (P_i). We call ((m_1,ldots,m_{n+1})) the degree of (mathcal{X}). For these polynomial vector fields (mathcal{X}) and in function of their degree we provide upper bounds, first for the maximal number of invariant (n)-dimensional spheres, and second for the maximal number of (n)-dimensional concentric invariant spheres.